Allison played collegiate football at the University of Memphis, which at the time was called Memphis State University. In 1992, his junior season at the school, Allison won the Lou Groza Award, awarded to the nation's top placekicker. During the season, Allison made 23 out of 25 field goals, and led the NCAA in both field goals made as well as and field goal percentage, where he achieved 92 percent. He also connected on all 32 extra points that he attempted. For the 1992 season, Allison scored a total of 101 points. Allison was also named by the Associated Press to the first-team All-American team, the first Memphis State University player in the school's history to be honored with that recognition.
Joe Allison's first cousin was Davey Allison, who was a NASCARrace car driver for Robert Yates Racing. Davey was killed in a helicopter accident in 1993. Joe was the godfather to Davey's daughter Krista. Joe and Davey were very close, and after his death, Joe said "Before I go in to kick, I put my hat down and say, 'Davey, give me the strength to do the best I can, to be as strong as you were.'" Joe is currently a special education teacher at North Forsyth High School in Cumming, Georgia.